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First, I don't think it's fair to characterize it as a lie. Second, I think Godel's Incompleteness Theorem shows that there are arguments that not everything i

First, I don't think it's fair to characterize it as a lie. Second, I think Godel's Incompleteness Theorem shows that there are arguments that not everything in math can be described by sets. [1] https://math.stackexchange.com/questions/2674701/how-why-does-g%C3%B6dels-incompletness-theorem-apply-to-set-theory https://math.stackexchange.com/questions/2674701/how-why-doe...

constantcrying 3y ago
>Second, I think Godel's Incompleteness Theorem shows that there are arguments that not everything in math can be described by sets. It doesn't. In fact it has no relation to what is and isn't a set. The problem is that "can be described" is not a mathematical question* and neither is "set" a universally defined term. You definitely can define a mathematical object which isn't a set in ZFC, e.g. {x: x is a set}, which is a universe ( https://en.m.wikipedia.org/wiki/Universe_(mathematics) https://en.m.wikipedia.org/wiki/Universe_(mathematics) ), but obviously it is not a set in ZFC. Notoriously {x: x is a set} is a set in naive set theory (which consequently is an incoherent theory). In a broad sense Category Theory considers objects which can not be sets under certain set-theories, but in a meta-mathematical way that is pretty much the point. Of course the correct point the author is trying to make is that standard mathematics is founded on set theory and is essential to almost every area you study in mathematics. "2" is a set "f:R \to R, x -> x^2" is a set and so on. *: Actually it is a mathematical question (see representation theory) but only makes sense if you ask whether something is representable as a specific set theory, which is not a precisely defined domain. N is a set in ZFC (or ZF-C) but definitely not in ZF-I, so "which set theory" is the real question.
- webnrrd2k 3y ago
  Re: "not everything in math can be described by sets” This is probably a better example: Russel's Paradox. Does there exist a set of all sets that do not contain themselves? I think it's a clear example of something that can't be described by standard set theory. "Set Theory", without further qualification, I think that it's safe to assume that they are talking about ZFC (or maybe Naive) set theory. Also, this is a reply to a post, so I need to make some assumptions about the audience, while keeping the post brief and coherent. I'm not trying to open up a discussion on what statements are true in every possible version of set theory, or the implication in other branches of math.
  - constantcrying 3y ago
    >I think it's a clear example of something that can't be described by standard set theory. It absolutely can be described by naive set theory. You literally just did it. It also is not a mathematical object in ZFC, there is no way to construct it. >I'm not trying to open up a discussion on what statements are true in every possible version of set theory, or the implication in other branches of math. But that is the only way the statement could ever be false. In standard mathematics, founded on ZFC, every object is a set. So the only way the statement could ever be false is if you look at different foundations and broader theories.
downvotetruth 3y ago
It is an intentionally false statement stated without qualification that it should not be taken as true. It is not ignorance as the author knows what (logical) qualifications are as the statement itself starts with the qualification "All" and the statement alone is false with no preceding context. To not make it false, it could be further qualified with "Almost" at the start, or use "Most" instead of "All".

constantcrying 3y ago

>Second, I think Godel's Incompleteness Theorem shows that there are arguments that not everything in math can be described by sets. It doesn't. In fact it has no relation to what is and isn't a set. The problem is that "can be described" is not a mathematical question* and neither is "set" a universally defined term. You definitely can define a mathematical object which isn't a set in ZFC, e.g. {x: x is a set}, which is a universe ( https://en.m.wikipedia.org/wiki/Universe_(mathematics) https://en.m.wikipedia.org/wiki/Universe_(mathematics) ), but obviously it is not a set in ZFC. Notoriously {x: x is a set} is a set in naive set theory (which consequently is an incoherent theory). In a broad sense Category Theory considers objects which can not be sets under certain set-theories, but in a meta-mathematical way that is pretty much the point. Of course the correct point the author is trying to make is that standard mathematics is founded on set theory and is essential to almost every area you study in mathematics. "2" is a set "f:R \to R, x -> x^2" is a set and so on. *: Actually it is a mathematical question (see representation theory) but only makes sense if you ask whether something is representable as a specific set theory, which is not a precisely defined domain. N is a set in ZFC (or ZF-C) but definitely not in ZF-I, so "which set theory" is the real question.

webnrrd2k 3y ago
Re: "not everything in math can be described by sets” This is probably a better example: Russel's Paradox. Does there exist a set of all sets that do not contain themselves? I think it's a clear example of something that can't be described by standard set theory. "Set Theory", without further qualification, I think that it's safe to assume that they are talking about ZFC (or maybe Naive) set theory. Also, this is a reply to a post, so I need to make some assumptions about the audience, while keeping the post brief and coherent. I'm not trying to open up a discussion on what statements are true in every possible version of set theory, or the implication in other branches of math.
- constantcrying 3y ago
  >I think it's a clear example of something that can't be described by standard set theory. It absolutely can be described by naive set theory. You literally just did it. It also is not a mathematical object in ZFC, there is no way to construct it. >I'm not trying to open up a discussion on what statements are true in every possible version of set theory, or the implication in other branches of math. But that is the only way the statement could ever be false. In standard mathematics, founded on ZFC, every object is a set. So the only way the statement could ever be false is if you look at different foundations and broader theories.

downvotetruth 3y ago

It is an intentionally false statement stated without qualification that it should not be taken as true. It is not ignorance as the author knows what (logical) qualifications are as the statement itself starts with the qualification "All" and the statement alone is false with no preceding context. To not make it false, it could be further qualified with "Almost" at the start, or use "Most" instead of "All".